How to get the partial factor representation of $\frac{1}{x^2+4x+1}$?

Question

I've been trying to factor this, but I don't see it.

What is the partial fractions representation of $$\frac{1}{x^2+4x+1}$$?

Answer 1

Yes, it does, but not over the rationals. You can use the quadratic formula or complete the square to say $x^2+4x+1=x^2+4x+4-3=(x+2+\sqrt 3)(x+2-\sqrt 3)$ and the denominators of the partial fractions are those two factors.

Answer 2

Yes. Let $\alpha, \beta$ be the roots of $x^2+4x+1$. Then $$\frac{1}{x^2+4x+1} = \frac{1}{(x-\alpha)(x-\beta)} = \frac{1/(\beta-\alpha)}{x-\beta}+\frac{1/(\alpha-\beta)}{x-\alpha}$$